Jaynes–Cummings–Hubbard model


The Jaynes-Cummings-Hubbard model is a many-body quantum system modeling the quantum phase transition of light. As the name suggests, the Jayne-Cummings-Hubbard model is a variant on the Jaynes–Cummings model; a one-dimensional JCH model consists of a chain of N coupled single-mode cavities, each with a two-level atom. Unlike in the competing Bose-Hubbard model, Jayne-Cummings-Hubbard dynamics depend on photonic and atomic degrees of freedom and hence require strong-coupling theory for treatment. One method for realizing an experimental model of the system uses circularly-linked superconducting qubits.
. In the circle, photon emission and absorption are shown.

History

The JCH model was originally proposed in June 2006 in the context of Mott transitions for strongly interacting photons in coupled cavity arrays. A different interaction scheme was synchronically suggested, wherein four level atoms interacted with external fields, leading to polaritons with strongly correlated dynamics.

Properties

Using mean-field theory to predict the phase diagram of the JCH model, the JCH model should exhibit Mott insulator and superfluid phases.

Hamiltonian

The Hamiltonian of the JCH model is
:
where are Pauli operators for the two-level atom at the
n-th cavity. The is the tunneling rate between neighboring cavities, and is the vacuum Rabi frequency which characterizes to the photon-atom interaction strength. The cavity frequency is and atomic transition frequency is. The cavities are treated as periodic, so that the cavity labelled by n = N+1 corresponds to the cavity n = 1. Note that the model exhibits quantum tunneling; this is process is similar to the Josephson effect.
Defining the photonic and atomic excitation number operators as and, the total number of excitations a conserved quantity,
i.e.,.

Two-polariton bound states

The JCH Hamiltonian supports two-polariton bound states when the photon-atom interaction is sufficiently strong. In particular, the two polaritons associated with the bound states exhibit a strong correlation such that they stay close to each other in position space. This process is similar to the formation of a bound pair of repulsive bosonic atoms in an optical lattice.